Elementary Magma Gradings on Rings

TL;DR

This paper generalizes the classification of gradings on rings by establishing a bijection between elementary magma gradings and magma homomorphisms, extending previous group-based results to magma and category graded rings.

Contribution

It introduces a bijection between elementary magma gradings and magma homomorphisms, broadening the scope from groups to magmas and categories, and applies this to determine grading cardinalities.

Findings

Bijection between elementary $H$-gradings and magma homomorphisms from $G$ to $H

Isomorphism between elementary $H$-filters and submagmas of $G imes H$

Application to category graded rings and groupoids, including cardinality results

Abstract

Suppose that $G$ and $H$ are magmas and that $R$ is a strongly $G$-graded ring. We show that there is a bijection between the set of elementary (nonzero) $H$-gradings of $R$ and the set of (zero) magma homomorphisms from $G$ to $H$. Thereby we generalize a result by D\u{a}sc\u{a}lescu, N\u{a}st\u{a}sescu and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of elementary (nonzero) $H$-filters on $R$ and the preordered set of (zero) submagmas of $G \times H$. These results are applied to category graded rings and, in particular, to the case when $G$ and $H$ are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of elementary $H$-gradings on $R$.